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Algebra II Name:_________________ Hr:_____ Semester 2 Final Exam REVIEW 2014 CHAPTER 12 – Trigonometry Evaluate the six trig functions of the angle . 1. sin 9 = csc = cos = sec = tan = cot = 12 Find the value of x for the triangle. Round your answers to the nearest hundredth. 2. 3. 4. 35° 4 9 3 63° 23° x x Find the EXACT value of x and y for the triangle. HINT: These are SPECIAL TRIANGLES. 5. 6. x 60° y 3 3 45° x y x = _________ , y =__________ Find the value of angle 7. θ 5 x = _________ , y =__________ for the triangle. Round answers to the nearest hundredth. 8. 17 11 θ 12 PAGE 1 x 9. You are standing on an observation deck 550 feet from the base of Mt. Rushmore. You look up at the top of Mount Rushmore at an angle of 39°. How high is the top of the monument? Round your answer to the nearest hundredth. 10. A flagpole casts a shadow 10 feet long. Viewed from the end of the shadow, the top of the flagpole makes a 63° angle with the ground. Sketch a diagram that represents this situation. What is the height of the flagpole? Round your answer to the nearest hundredth. Draw the following angles in standard position. 11. 150 12. 480 13. -45 Convert the following angles to radians. (Use: ___________) 14. 80 15. 120 16. -135 Convert the following angles to degrees. (Use: ___________) 7 17. 18. 6 4 Find one positive and one negative angle that are coterminal with the angle given. 3 20. 50 21. -200 22. 4 PAGE 2 19. 2 3 23. 3 Use the given point on the terminal side of an angle in standard position to evaluate sine, cosine, and tangent. 24. (7, 24) 25. (-3, 4) sin = sin = cos = cos = tan = tan = Sketch the angle. Then find its reference angle. 26. 150° 27. 240° 28. 3 4 Evaluate the function without using a calculator. HINT: Find the reference angle and then use SPECIAL TRIANGLES to evaluate. 11 29. sin 120° 30. cos 210° 31. tan 6 To solve triangles, we learned how to use two different laws: LAW OF SINES: o Use when you have ___ ___ ___ or ___ ___ ___ LAW OF COSINES: o Use when you have ___ ___ ___ or ___ ___ ___ PAGE 3 32. If A = 40°, C = 75°, c = 20, find the length of side b. 33. If A = 52°, a = 32, b = 42, find the measure of angle B. 34. If B = 70°, b = 85, c = 88, find the measure of angle C. 35. If C = 115 , a = 4, b = 6, find the length of side c. 36. If a = 16, b = 14, c = 18, find the measure of the largest angle. 37. Two airplanes leave an airport at the same time, and the angle between their flight paths is 40º. An hour later, one plane has traveled 300 miles while the other has traveled 200 miles. How far apart are the planes at this time? Round your answer to the nearest hundredth. 38. Two forest rangers, 12 miles from each other on a straight service road, both site an illegal bonfire off the road. Using their radios to communicate, they discover the fire is between them. The first ranger’s line of sight to the fire makes an angle of 38° with the road and the second ranger’s line of sight to the fire makes a 63° angle with the road. How far is the fire from each ranger? Round your answer to the nearest hundredth. PAGE 4 CHAPTER 12 ANSWERS 3 3 4 5 1. sin: cos: tan: csc: 5 5 3 4 sec: 3 2 3 2 , y 2 2 4. x = 6.61 5. x 9. 10. 19.63 feet 14. 445.38 feet 4 9 15. 2 3 21. 160°, -560° 25. sin = 30. 16. 3 4 22. 495°, -225° 4 3 4 cos = tan = 5 5 3 3 2 5 4 cot: 3 4 31. 34. C = 77° OR 103° 3 3 35. c = 8.5 2. x = 21.20 3. x = 2.29 6. x 3 3 , y 6 11. 7. = 27.04° 12. 17. 315° 18. 30° 23. 300°, -420° 26. 30° 32. b = 18.8 36. C = 73° = 54.78° 13. 19. -120° 24. sin = 27. 60° 8. 20. 410°, -310° 24 7 24 cos = tan = 25 25 7 28. 45° 29. 3 2 33. No Solution 37. 195.13 miles 38. 10.89 miles and 7.53 miles Statistics and Probability Find the mean, median, and range of the data set. Round answers to the nearest tenth. 1. 3, 3, 4, 5, 5, 7, 8, 10, 11 2. 95, 76, 88, 82, 73, 65, 76, 76, 84, 90 Mean:_________________ Mean:_________________ Median:_______________ Median:_______________ Range:________________ Range:________________ Find the median, upper quartile, lower quartile, maximum/upper extreme, and minimum/lower extreme. Then construct a box-and-whisker plot of the data. 3. 25, 56, 43, 44, 35, 31, 73, 66, 62, 29, 37 Median:_______ Upper Quartile: _______ Lower Quartile: _______ Maximum: _______ Minimum: _______ PAGE 5 4. 17, 38, 22, 15, 13, 24, 18, 10, 20, 13, 17, 12 Median:_______ Upper Quartile: _______ Lower Quartile: _______ Maximum: _______ Minimum: _______ Given the probability of a certain event, state the odds of that event 5. 3 5 6. 1 4 7. Given the odds of a certain event, state the probability of that event 8. 5:16 9. 4:21 10 27 10. 12:7 Given the spinner below, state the probability of each outcome. 11. Landing on red:_________________ 12. Landing on orange or purple:_______________ 13. Does not land on green:________________ 14. Does not land on purple or orange:________________ 15. Does not land on purple or orange or red or green:____________ A jar contains 3 white, 1 red, 6 green and 5 blue marbles. If one marble is drawn at random, state the probability of each outcome. 16. A white marble 17. A red marble 18. A black marble 19. Not a white marble 20. Not a blue marble 21. A red or green marble PAGE 6 Given a standard deck of 52 cards, if one card is drawn at random, state the probability and odds of each situation. 22. A black card 23. A ten 24. A red jack 25. Not a black card 26. Not a four 27. Not a black jack 28. A men’s department store sells 3 different suit jackets, 6 different shirts, 8 different ties, and 4 different pairs of pants. How many different suits consisting of a jacket, shirt, tie, and pants are possible? 29. Subway offers 5 different types of bread, 10 different types of meat, 3 different types of cheese, 9 different vegetable toppings, and 7 different condiments. If you can pick only one type of bread, one meat, one cheese, one veggie, and one condiment, then how many different subs can you order? 30. A license plate will consist of three letters followed by two digits. How many different license plates are possible? 31. A license plate will consist of 2 letters and 4 digits and no letters or digits can be repeated. How many license plates are possible? Evaluate each permutation or combination. 32. 4 P3 33. 8 P4 34. 6 C4 35. 14 C10 36. A photographer lines up the 13 players of a basketball team in a single line to take a team picture. How many different ways can the photographer arrange the team for the picture? 37. A teacher must choose 4 students from the 20 students in your chemistry class to represent your school in an Academic Challenge. How many different combinations of 4 students can the teacher choose? PAGE 7 38. Nine students enter a race. Awards are given for first place through third place. In how many ways can the students finish first through third? 39. A teacher is holding tryouts for the school play. There are 15 students trying out for 7 parts. Each student can play each part. In how many ways can the teacher select the students? 40. A pizza parlor offers ten different toppings. How many different 5-topping pizzas can be created using the 10 toppings? (Assume no topping is used more than once.) STATISTICS AND PROBABILITY ANSWERS 1. Mean: 6.2; Median: 5; ; Range: 8 3. Median: 43, UQ: 62, LQ: 31, Max: 73, Min: 25 5. 3:2 6. 1:3 2 5 13. 4 5 14. 3 5 19. 4 5 20. 2 3 21. 7 15 1 Odds: 1:25 26 25. Prob: Mean: 80.5; Median: 79; Range: 30 4. 5 8. 21 1 15. 5 7. 10:17 12. 24. Prob: 2. Median: 17, UQ: 21, LQ: 13, Max: 38, Min: 10 12 4 1 9. 10. 11. 19 25 5 1 0 1 16. 17. 18. 15 15 5 1 1 Odds: 1:1 23. Prob: Odds: 1:12 13 2 12 25 26. Prob: Odds: 12:1 27. Prob: Odds: 25:1 13 26 22. Prob: 1 Odds: 1:1 2 28. 576 29. 9,450 30. 1,757,600 31. 3,276,000 32. 24 33. 1,680 34. 15 35. 1,001 36. 6,227,020,800 37. 4,845 38. 504 39. 6,435 40. 252 PAGE 8 CHAPTER 9 – Rational Functions Graph each of the following functions and fill in the missing information. 1. y 2 x3 x y x y Vertical Asymptote: Horizontal Asymptote: Domain: Range: x-intercept: y-intercept: 2. y 1 1 x2 Vertical Asymptote: Horizontal Asymptote: Domain: Range: x-intercept: y-intercept: Simplify each expression. 3. 6x 3 4x 2 1 4 6. 4. 4 y 2 27 9 x 16 xy2 7. 2 x 4 x x 3x 2 x2 4 3x 6 2 32 x y 8 xy 3 xy2 21 y 4 x 2 2 x 15 x 2 x 30 9. x 2 11x 24 x 2 3 x 18 2 10. 5. 2 2 3 2 3x 2 x n4 8. n 2 6n 9 n 2n 8 3 n 2 11. PAGE 9 x 2 3x x2 x 2 x 2 3x 2 x 2 4 x 3 4 5 3 2 6x 3x 12. x 3 x4 x2 13. 3x 2 x4 x5 14. x 3x 2 x 1 x 1 15. 3x 4x 2 x2 x 4 16. 3 2 x 8 x 15 x 5 17. x 2x 2 2 3x 15 x 4 x 5 2 Solve the following equations. 18. x 2x 2x 1 x 2 19. 20. 7 3 3 2 x 21. 8 PAGE 10 15 4 7 x 5 x 2 2x x 1 x 1 22. 2 3 2 x 1 x 24. x 1 5 2 x 1 x 1 x 1 23. 5 1 5 x2 x x CHAPTER 9 ANSWERS 1. 3. 8. 13. VA: x = -3 HA: y = 0 D: ARN except -3 R: ARN except 0 x-int: None y-int: (0, 2/3) 3 3 4. 2x 1 4x 2 1 n 3n 2 3x 2 13 x 8 x 4x 5 9. 14. x 3 x 8 x x 4 x 1x 1 2. VA: x = -2 HA: y = -1 D: ARN except -2 R: ARN except - 1 x-int: (-1, 0) y-int: (0, -1/2) 7. 6x x 2x 1 2x 5 3x 3 12. x 2 x 12 x 4x 2 2 x 3 x 5x 3 17. x6 3x 5 5. x x 1 6. 28x 2 y 10. 4x 9 11. 16. 15. 6x 2 x3x 10 x 2x 2 18. x = 0 19. x = -10 20. x = -6 23. x = 8 24. x = 3, 2 25. x = 3 2 PAGE 11 21. No Solution 22. x = ½, 3 Chapter 11 – Sequences and Series Write the first five terms of the sequence. 1. a n n 1 2. an 3n 5 3. an n 2 6 Determine whether the following sequences are arithmetic, geometric, or neither. Then find the next term. 4. 1, 4, 7, 10, 13,… 5. 1, -2, 3, -4, 5, … 6. 2, 4, 8, 16, 32 Write a rule for the nth term of the arithmetic sequence. Then find a 8. 7. 7, 10, 13, 16, … 8. 3, 1, -1, -3, -5,… 9. d = 2, a5 = 1 10. d = -5, a10 = -60 Find the sum of the first 10 terms of the arithmetic series. 11. -1 + 2 + 5 + 8 + 11 +… 12. 4 + 3 + 2 + 1 + 0 + … 13. 3 + 6 + 9 + 12 + 15 +… Write a rule for the nth term of the geometric sequence. Then find a6. 14. 4, 8, 16, 32,… 15. 5000, 500, 50, 5,… 16. r = 3, a1 = 4 17. r = 2, a1 = 1 Find the sum of the first 10 terms of the geometric series. 18. 2 + 4 + 8 + 16 + 32 +… 19. 1 + 3 + 9 + 27 + … PAGE 12 20. ½ + 1 + 2 + 4 + … CHAPTER 11 ANSWERS 1. 2, 3, 4, 5, 6, 7 2. -2, 1, 4, 7, 10, 13 3. 7, 10, 15, 22, 31, 42 4. Arithmetic, 16 5. Neither, -6 6. Geometric, 64 7. an = 3n + 4; a8 = 28 8. an = -2n + 5; a8 = -11 9. an = 2n - 9; a8 = 7 10. an = -5n – 10; a8 = -50 14. an = 4(2)n-1; a6 = 128 17. an = 1(2)n-1; a6 = 32 11. 125 12. -5 1 1 15. an = 5000 n-1; a6 = 0.05 = 20 10 13. 165 18. 2046 20. 511.5 19. 29,524 16. an = 4(3)n-1; a6 = 972 Chapter 13 – Conic Sections Identify the center and radius of the circle. Then graph the circle. 1. x 2 y 2 25 2. x 2 y 2 10 3. x 32 y 12 9 Center:______________ Center:______________ Center:______________ Radius:______________ Radius:______________ Radius:______________ Write the equation of a circle that satisfies the given information. 4. Center (0, 0) r = 4 5. Center (-5, 7) r = 2 3 PAGE 13 6. Center (0, 0); point on circle (2, -6) Graph the ellipse. Identify the vertices, co-vertices, and foci. 7. x2 y2 1 25 4 8. x2 y2 1 9 36 9. 9 x 2 16 y 2 144 Vertices:_______________ Vertices:______________ Vertices:_______________ Co-Vertices:___________ Co-Vertices:___________ Co-Vertices:___________ Foci:__________ Foci:__________ Foci:__________ Write an equation for the ellipse with center at (0, 0) that satisfies the given information. 10. Vertex (0, -4) Co-vertex (3, 0) 11. Vertex (5, 0) Focus (4, 0) Identify the focus and directrix for the parabola and graph the equation. 12. y 2 12 x 13. x2 8y Focus:_______________ Focus:_______________ Directrix:_____________ Directrix:_____________ PAGE 14 14. x 2 4 y 15. y 2 16 x Focus:_______________ Focus:_______________ Directrix:_____________ Directrix:_____________ Write the standard form of the equation of the parabola with vertex at (0, 0) and the given focus or directrix. 16. Directrix: x = -3 17. Directrix: y = 4 18. Focus: (0, 5) 19. Focus: ( 1 , 0) 2 Graph the hyperbola. Identify the vertices, foci, and asymptotes. 20. x2 y2 1 25 4 21. y2 x2 1 16 9 22. y 2 16 x 2 16 Vertices:_____________________ Vertices:_____________________ Vertices:_____________________ Foci:_________________________ Foci:_________________________ Foci:_________________________ Asymptotes:__________________ Asymptotes:__________________ Asymptotes:__________________ PAGE 15 Write an equation of the hyperbola centered at (0, 0) with the given foci and vertices. 23. Foci: (0, -4) (0, 4) Vertices: (0, -2) (0, 2) 24. Foci: (-6, 0) (6, 0) Vertices: (-3, 0) (3, 0) CHAPTER 13 ANSWERS 1. C(0, 0); r = 5 5. x 52 y 7 2 12 8. V (0, 11. x2 y2 1 25 9 y 2 2 x 21. V (0, 23. 6. 12. F(3, 0) dir. x = -3 16. x2 y2 1 9 27 3); F( 7 , 0) 13. F(0, 2); y = -2 y 2 12 x 20. V ( 5, 0); F( 24. 7. V ( 5, 0); CV(0, 9. V ( 4, 0); CV(0, 4 4); F(0, 5); Asymp. y x 3 y2 x2 1 4 12 3. C(3, -1); r = 3 10 x 2 y 2 40 6); CV( 3, 0); F(0, 27 ) 15. F(-4, 0); x = 4 19. 2. C(0, 0); r = 17. x 2 16 y 4. x 2 y 2 16 2); F( 21 , 0) 10. x2 y2 1 9 16 14. F(0, -1); y = 1 18. x 2 20 y 2 29 , 0); Asymp. y x 5 22. V(0, 4); F(0, 17 ); Asymp. y 4 x CHECK ONLINE KEY FOR GRAPHS!!! PAGE 16